Are many shops/restaurants/guesthouses shut down, or is it business as usual? Are there many roadblocks set up in Bangkok and around the country, which may potentially slow a traveler down?
Currently it is business as usual here in Thailand. Guesthouses, hotels, restaurants, shops, buses, planes, etc are all operating as normal. International flights are coming and going.
The Army has imposed no restrictions on travel
There are army checkpoints scattered around Bangkok, which may increase drive times. But they are primarily looking for weapons and protest supplies, so taxis full of tourist get passed through pretty quickly. Both airports in Bangkok (Suvarnabhumi and Don Muang) are advising leaving earlier if you have a flight to catch, perhaps 3 to 4 hours ahead of departure, just in case traffic is heavy when you reach a checkpoint.
Outside of Bangkok you wouldn't know we were under martial law if you didn't see it in the news.
Shortly after I posted my reply, the Army declared a coup d'etat. The Army has imposed a 10pm to 5am curfew, schools are closed today & tomorrow (23/24 May) and TV channels are only carrying official announcements.
But the majority of my answer still applies, as tourist services, shops, businesses are operating as usual. Travelers with flights during curfew periods are still allowed to travel to and from the airports.
The situation there is fluid... even if someone can say "nothing is closed" (and I'm sure there are disruptions right now because there are protester encampments, and right now the army is stepping in which is not a great sign) the situation can completely change in 24-48 hours depending on events.
Typically in their political crises most of the disruption is confined to some parts of Bangkok (and usually the other parts are business as usual).
I was in Thailand late June to early July 2014 (Hat Yai, Songkhla, Khanom, Ko Samui, Ko Phangan, Ko Tao, Bangkok, Ayutthaya). By then, the curfew had already been completely lifted nationwide.
As a tourist doing the usual touristy stuff, there was no sign whatsoever of any political unrest or military coup. I experienced no inconveniences.
At present it appears that the situation has stabilized and will remain so for at least the coming months. (The military promises elections only in October 2015.)